Title Perturbative Expansion of the Chern-Simons Integral(Mathematical Quantum Field Theory and Related Topics)

Author(s) Mitoma, Itaru

Citation 数理解析研究所講究録 (2013), 1859: 9-30

Issue Date 2013-10

URL http://hdl.handle.net/2433/195285

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

Perturbative Expansion of the Chern-SimonsIntegral

Itaru Mitoma

Department of Mathematics, Saga University

1 Introduction

The necessity of the asymptotic expansion in infinite dimensions arises fromthe pioneering work of the Chern-Simons theory by Witten [10] in 1989.

However, the exponentia13rd term of the Chern-Simons theory is less likelyto be handled by techniques known at present, so that we will challenge a newmethod called the Fujiwara-Kumano-go method [5, 8] as a possibility.

Let $M$ be a compact oriented smooth 3-manifold, $G$ a simply connected,connected compact simple Lie group, and $Parrow M$ a principal $G$-bundle over$M$ . Let denote by $\Omega^{r}(M, \mathfrak{g})$ , the space of $\mathfrak{g}$-valued smooth $r$-forms on $M.$

Let $\mathcal{A}$ denote the space of connections on $P$ and $\mathcal{G}$ the group of gaugetransformations on $P$ . Note that, by fixing a reference connection on $P$ as theorigin, we may identify $\mathcal{A}$ with the (infinite-dimensional) vector space $\Omega^{1}(M, \mathfrak{g})$ ,and $\mathcal{G}$ with the space $C^{\infty}(M, G)$ of smooth maps from $M$ to $G$ , respectively.Then the Chem-Simons integral of an integrand $F(A)$ is given by

(1.1) $\int_{A/\mathcal{G}}F(A)e^{L(A)}\mathcal{D}(A)$ ,

where the Chern-Simons Lagrangian $L$ is defined by

(1.2) $L(A)=- \frac{ik}{4\pi}\int_{M}$ Tr $\{A\wedge dA+\frac{2}{3}A\wedge A\wedge A\},$

$\mathcal{D}(A)$ is the Feynman measure and the parameter $k$ is a positive integer calledthe level of charges. $($

Among various integrands, the most typical example of gauge invariantobservables is the Wilson line defined by

(1.3) $F(A)= \prod_{j=1}^{s}Tr_{R_{j}}\mathcal{P}\exp l_{j}A,$

where $\mathcal{P}$ denotes the product integral (see [4]), $\gamma_{j},$ $j=1,2,$ $\ldots,$$s$ , are closed

oriented loops, and the trace $\ulcorner\Gamma r$ is taken with respect to some irreduciblerepresentation $R_{j}$ of $G$ assigned to each $\gamma_{j}.$

数理解析研究所講究録第 1859巻 2013年 9-30 9

In Section 2, we give several relevant basic notions, after which we stateour results precisely. In Section 3, we prove theorems stated in Section 2 byusing the Fujiwara-Kumano-go method.

Throughout this paper, $\sqrt{z}$ is understood to denote the branch of the rootof $z\in C$ where $- \frac{\pi}{2}<arg\sqrt{z}<\frac{\pi}{2}.$

2 Definitions and Results

From the method of superfields of the perturbative formulation of theChern-Simons integral [2, 3], we have the Lorentz gauge fixed form of theChern-Simons integral written as

(2.1)$\int_{A}\int_{\Phi}\int_{\hat{C}},$ $\int_{C},$ $\mathcal{D}(A)\mathcal{D}(\phi)\mathcal{D}(\tilde{c})\mathcal{D}(c’)F(A_{0}+A)$

$\cross\exp[ik((A, \phi), Q_{A_{0}}(A, \phi))_{+}-\frac{ik}{4\pi}\int_{M}Tr\frac{2}{3}A\wedge A\wedge A+\int_{M}$ Tr $\tilde{c}d_{A_{0}}*D_{A}c’].$

Here $A_{0}$ is a background connection, $Q_{A_{0}}$ is a twisted Dirac operator and$(,$ $)_{+}$ is the inner product of the Hilbert space $L^{2}(\Omega_{+})=L^{2}(\Omega^{1}(M, \mathfrak{g})\oplus\Omega^{3}(M, \mathfrak{g}))$

given by$((A, \phi), (B, \varphi))_{+}=(A, B)+(\phi, \varphi)$ ,

where the inner product and the norm on $\Omega^{r}(M, \mathfrak{g})$ are defined by

(2.2) $( \omega, \eta)=-\int_{M}Tx\omega\wedge*\eta, |\cdot|=\sqrt{(}, )$ .

By heuristically considering

$\int_{\hat{C}},$ $\int_{C},$ $\mathcal{D}(\tilde{c})\mathcal{D}(c’)\exp[\int_{M}$ Tr $\tilde{c}d_{A_{0}}*D_{A}c’]$

$=\det*d_{A_{0}}*D_{A}=\det\Delta_{0}\det_{R}*d_{A_{O}}*D_{A},$

and balancing out $\det\Delta_{0}$ by a normalizaion of (2.1), we arrive at the perturba-tive heuristic formulation of the normalized Chern-Simons integral such that

$\frac{1}{Z}\int_{A}\int_{\Phi}\mathcal{D}(A)\mathcal{D}(\phi)F(A_{0}+A)$

(2.3)$\cross\det_{R}*d_{A_{0}}*D_{A}\exp[ik((A, \phi), Q_{A_{0}}(A, \phi))_{+}-\frac{ik}{4\pi}\int_{M}Tr\frac{2}{3}A\wedge A\wedge A],$

where

(2.4) $Z= \int_{A}\int_{\Phi}\mathcal{D}(A)\mathcal{D}(\phi)\exp[ik((A, \phi), Q_{A_{0}}(A, \phi))_{+}].$

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To provide mathematical meaning to this, we have first of all to regularizethe formal determinant $\det_{R}*d_{A_{0}}*D_{A}$ . Briefly we recall Albeverio-Mitoma[l].

Since $Q_{A_{0}}$ is self-adjoint and elliptic, $Q_{A_{0}}$ has pure point spectrum [8]. Let$\lambda_{j},$ $e_{j}=(e_{j}^{A}, e_{j}^{\phi}),$ $j=1,2,$ $\cdots$ be the eigenvalues and vectors of $Q_{A_{0}}$ in $L^{2}(\Omega_{+})$ .

Let $\{\nu_{j}, \xi_{j}, j=1,2, \cdots\}$ be the eigensystem of $\triangle 0$ in $L^{2}(\Omega^{0})$ ,

$\tilde{c}=\sum_{j=1}^{\infty}\xi_{j}\hat{c}_{j}’$ and $c’= \sum_{j=1}^{\infty}\xi_{j}c_{j}’.$

Define

$a_{R,ij}^{\ell}=- \frac{1}{\nu_{i}}\int_{M^{r}}\Gamma rr_{i}d_{A_{0}}\xi_{i}\wedge*[e_{\ell}^{A}, r_{j}\xi_{j}].$

Since $M$ is compact, we can choose appropriate real numbers $r_{j}>0,j=$$1,2,$ $\ldots$ such that

(2.5)$\sum_{i,j}|a_{R,ij}^{\ell}|<1.$

For an integer $p$ , we define the Hilbert subspace $H_{p}(\Omega_{+})$ of $L^{2}(\Omega_{+})$ with newinner product $()_{p}$ defined by

$((A, \phi), (B, \varphi))_{p}=(A, (I+Q_{A_{0}}^{2})^{p}B)+(\phi, (I+Q_{A_{0}}^{2})^{p}\varphi)$ ,

where $I$ is the identity operator on $L^{2}(\Omega_{+})$ , and the $p-$-norm on $H_{p}(\Omega_{+})$ is definedas usual by $\Vert\cdot\Vert_{p}=\sqrt{(},$$)_{p}$ .Henceforth we denote $H_{p}(\Omega_{+})$ briefly by $H_{p}$ wheneverno confusion may occur. Let $h_{i}=(1+\lambda_{i}^{2})^{-p/2}e_{i}$ be the CONS of $H_{p}$ . Choose asufficiently large $p$ satisfying the condition

$\sum_{i=1}^{\infty}(1+\lambda_{i}^{2})^{-p}|\lambda_{i}|<\infty,$

and guaranteeing the regularizations in what follows.From now on, we use the brief notations such that

$\beta_{j}=(1+\lambda_{i}^{2})^{-p/2}$ and $a_{j}=\beta_{j}^{2}\lambda_{j}.$

Then for $(A, \phi)\epsilon H_{p}$ , we consider instead of $\det_{R}*d_{A_{0}}*D_{A}$ , the regularizeddeterminant defined by

$\det_{Reg}(A)=\lim_{marrow\infty}\det_{Reg}^{m}(A)$ ,

where

$\det_{Reg}^{m}(A)=$

$a_{11}^{R}(A)$ $a_{12}^{R}(A)$ $a_{1m}^{R}(A)$

$a_{21}^{R}(A\rangle$ $a_{22}^{R}(A)$ $a_{2m}^{R}(A)$

$a_{m1}^{R}(A)$ $a_{m2}^{R}(A)$ $a_{mm}^{R}(A)$

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$a_{ii}^{R}(A)=1+ \sum_{\ell=1}^{\infty}\beta_{\ell}((A, \phi), h_{\ell})_{p}a_{R,ii}^{\ell}$

and

$a_{ij}^{R}(A)= \sum_{\ell=1}^{\infty}\beta_{\ell}((A, \phi), h_{\ell})_{p}a_{R,ij}^{\ell}.$

The regularized determinant is well defined for sufficiently large $p$ , whichis guaranteed by the increasig rates of eigenvalues of $Q_{A_{0}}((c)$ of Lemma 1.6.3in [6] $)$ .

Next we proceed to a regularizing the holonomy. From Mitoma-Nishikawa [9],for a given closed smooth curve $\gamma$ : $[0,1]arrow M$ in $M$ , for each $t\in[O, 1]$ and suffi-ciently small $\epsilon>0$ there exists a Poincare dual $C_{\gamma}^{\epsilon}(t)$ such that

$\lim_{\epsilonarrow 0}\sup_{0\leq t\leq 1}|(A, C_{\gamma}^{\epsilon}(t))-\sum_{i=1}^{3}\int_{0}^{t}A_{i}^{\alpha}(\gamma(\tau))\dot{\gamma}^{i}(\tau)d\tau|=0.$

Since$(A, C_{\gamma}^{\epsilon}(t))=((A, \phi), (I+Q_{A_{0}}^{2})^{-p}(C_{\gamma}^{\epsilon}(t), 0))_{p},$

by setting

(2.6) $\tilde{C}_{\gamma}^{\epsilon}(t)=(I+Q_{A_{0}}^{2})^{-p}(C_{\gamma}^{\epsilon}(t), 0)$ ,

we define

(2.7) $A_{\gamma}^{\epsilon}(t)= \sum_{\alpha=1}^{d}((A, \phi), \v{C}_{\gamma}^{\epsilon}(t)^{\alpha}\otimes E_{\alpha})_{p}E_{\alpha},$

where $\v{C}_{\gamma}^{\epsilon}(t)=\sum\v{C}_{\gamma}^{\epsilon}(t)^{\alpha}\otimes E_{\alpha},$ $\{E_{\alpha}\}$ is a basis of $\mathfrak{g}$ , and define

$\overline{A}(t)=\int_{[0,t]}A.$

By Chen’s iterated integral [4], we define the $\epsilon$-regularization of the holon-omy by

(2.8) $W_{\gamma}^{\epsilon}(A)=I+ \sum_{r=1}^{\infty}W_{\gamma}^{\epsilon,r}(A)$ ,

where

$W_{\gamma}^{\epsilon,r}(A)= \int_{0}^{1}\int_{0}^{t_{1}}\cdots\int_{0}^{t_{r-1}}d(\overline{A}_{0}+A_{\gamma}^{\epsilon})(t_{1})d(\overline{A}_{0}+A_{\gamma}^{\epsilon})(t_{2})\cdots d(\overline{A}_{0}+A_{\gamma}^{\epsilon})(t_{r})$,

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and the $\epsilon$-regularized Wilson line by

(2.9) $F_{A_{0}}^{\epsilon}(A)= \prod_{j=1}^{s}Tr_{R_{j}}W_{\gamma_{j}}^{\epsilon}(A)$ ,

where the trace Tr is $taken\neg$ in the representation $R_{j}$ of $G$ assigned to each loop$\gamma_{j}.$

Thus adding regularizations of the determinant and holonomy to (2.3), wehave

$\frac{1}{Z}\int_{\mathcal{A}}\int_{\Phi}\mathcal{D}(A)\mathcal{D}(\phi)F_{A_{0}}^{\epsilon}(A)$

(2.10)$\cross\det(A)\exp Reg[ik((A, \phi), Q_{A_{0}}(A, \phi))_{+}-\frac{ik}{4\pi}\int_{M}Tr\frac{2}{3}A\wedge A\wedge A],$

which is a heuristic version of the normalized Chern-Simons integral.Let us exploit compensations in the numerator and denominator.(see [1]).

Then setting$\tilde{F}_{A_{0}}^{\epsilon}(A)=_{Reg}\det(A)F_{A_{0}}^{\epsilon}(A)$ ,

we have a heuristic form of (2.10) such that

$\frac{1}{\tilde{Z}}\int_{\mathcal{A}}\int_{\Phi}\mathcal{D}(A)\mathcal{D}(\phi)\tilde{F}_{A_{0}}^{\epsilon}(\frac{1}{\sqrt[3]{k}}A)$

(2.11)$\cross\exp[i\sqrt[3]{k}((A, \phi), Q_{A_{0}}(A, \phi))_{+}-\frac{i}{4\pi}\int_{M^{\ulcorner}}\Gamma r\frac{2}{3}A\wedge A\wedge A],$

where

(2.12) $\tilde{Z}=\int_{A}\int_{\Phi}\mathcal{D}(A)\mathcal{D}(\phi)\exp[i\sqrt[3]{k}((A, \phi), Q_{A_{0}}(A, \phi))_{+}].$

Based on the heuristic idea such that the asymptotic expansion up to theorder $2N$ of (2.11) may be equal to the asymptotic expansion up to the order$2N$ of

$\frac{1}{\tilde{Z}}\int_{A}\int_{\Phi}\mathcal{D}(A)\mathcal{D}(\phi)\tilde{F}_{A_{0}}^{\epsilon}(\frac{1}{\sqrt[3]{k}}A)$

(2.13) $\cross\exp[i\sqrt[3]{k}((A, \phi), Q_{A_{0}}(A, \phi))_{+}-\frac{i}{4\pi}\int_{M}Tr\frac{2}{3}A\wedge A\wedge A]$

$\cross\exp[-(\sum_{j=1}^{\infty}|((A, \phi), e_{j})_{+}|)^{2N+2}],$

from now on, we discuss the asymptotic expansion of (2.13).For $x=(A, \phi)\in H_{p},$

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$x= \sum_{j=1}^{\infty}(x, h_{j})_{p}h_{j},$

$(x, Q_{A_{0}}x)_{+}= \sum_{j=1}^{\infty}a_{j}(x, h_{j})_{p}^{2},$

and

$\frac{-1}{4\pi}\int_{M}Tr\frac{2}{3}A\wedge A\wedge A=a,b,1\sum_{\subset}^{\infty}(x, h_{a})_{p}(x, h_{b})_{p}(x, h_{c})_{p}\beta_{a}\beta_{b}\beta_{c}T_{abc},$

where $T_{abc}=- \int_{M}Tr\frac{1}{6\pi}e_{a}^{A}\wedge e_{b}^{A}\wedge e_{c}^{A}.$

By the idea of justifying the Feynman integral due to It\^o [7] such that theconvergent factors

$\exp[-\frac{(x,x)_{m}}{2n}]$ with $n>0$

implemented into the $m$-dimensional approximation of (2.13) and using theformula of changing variables, by setting $x=\sqrt{n}y$ , we have

$\frac{1}{\hat{Z}_{m,n}}\int_{R^{m}}\tilde{F}_{A_{0}}^{\epsilon}(\frac{\sqrt{n}}{\sqrt[3]{k}}y^{m})\exp[in\sqrt[3]{k}(y^{m}, Q_{A_{0}}y^{m})_{+}]$

(2.14)$\cross\exp[i\sum_{ab\subset 1}^{\infty}\sqrt{n}y_{a}\sqrt{n}y_{b}\sqrt{n}y_{c}\beta_{a}\beta_{b}\beta_{c}T_{abc}]$

$\exp[-\frac{(y,y)_{m}}{2}-(\sum_{j=1}^{m}\beta_{j}\sqrt{n}|y_{j}|)^{2N+2}]\frac{\nu_{m}(dy)}{(\sqrt{2\pi})^{m}},$

where $\nu_{m}(dy)$ is the $m$-dimensional Lebesgue measure, $y^{m}= \sum_{j=1}^{m}y_{j}h_{j}$ , and

$\hat{Z}_{m,n}=\int_{R^{m}}\exp[in\sqrt[3]{k}(y^{m}, Q_{A_{0}y^{m}})_{+}]\exp[-\frac{(y,y)_{m}}{2}]\frac{\nu_{m}(dy)}{(\sqrt{2\pi})^{m}}.$

Setting

$f_{n}^{L}(xx, \cdots, x_{1})=\tilde{F}_{A_{0}}^{\epsilon}(\frac{\sqrt{n}}{\sqrt[3]{k}}x^{L})$

$\cross\exp[i\sum_{abc=1}^{L}\sqrt{n}x_{a}\sqrt{n}x_{b}\sqrt{n}x_{c}\beta_{a}\beta_{b}\beta_{c}T_{abc}]$

$\exp[-(\sum_{j=1}^{L}\beta_{j}\sqrt{n}|x_{j}|)^{2N+2}],$

we give the definition of a perturbative Chern-Simons integral (2.13) by settingit as equal to

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(2.15)$\lim_{Larrow\infty}\lim_{narrow\infty}\frac{1}{\hat{Z}_{L,n}}\int_{R^{L}}f_{n}^{L}(x_{L}, x_{L-1}, \cdots, x_{1})\exp[in\sqrt[3]{k}(^{-}x^{L}, Q_{A_{0}}x^{L})_{+}]$

$\exp[-\frac{(x,x)_{L}}{2}]\frac{\nu_{L}(dx)}{(\sqrt{2\pi})^{L}}.$

Replace the eigenvalues $\lambda_{j}$ of $Q_{A_{0}}$ by $\lambda_{j}^{\zeta}$ for large $\zeta$ satisfying

Assumption 1. RENORMALIZATION,

$\sum_{j=1}^{\infty}j^{2(8N^{2}+12N+4)}\frac{1}{\sqrt{|\lambda_{j}|}}\leq c<+\infty,$

which is guaranteed by Lemma 1.6.3 in [6].Then we have

Theorem 1. Under the renormalization Assumption 1 and the assumption $T_{abc}\leq T<$

$+\infty$ , we have (2.15) is equal to

$\tilde{F}_{A_{0}}^{\epsilon}(0)$

$+ hmLarrow\infty\{\sum_{s=1}^{N}(\sum_{r=1}^{S}(\sum_{1\leq j_{1}<j_{2}<j_{3}<j_{r}\leq L}\ldots(_{m_{1},m_{2}},\cdots,\sum_{m_{r}\geq 1,m_{1}+m_{2}+\cdots+m_{r}=s}$

(2.16)$n arrow\infty hm((\prod_{q=1}^{r}\frac{1}{2^{m_{q}}m_{q}!(1-2in\sqrt[3]{k}\beta_{j_{q}}^{2}\lambda_{j_{q}})^{m_{q}}}\partial_{x_{j_{q}}}^{2m_{q}})\tilde{F}_{A_{0}}^{\epsilon}(\frac{\sqrt{n}}{\sqrt[3]{k}}x^{L})$

$\cross\exp[i\sum_{abc=1}^{L}\sqrt{n}x_{a}\sqrt{n}x_{b}\sqrt{n}x_{c}\beta_{a}\beta_{b}\beta_{c}T_{abc}])(0))))\}$

$+O(( \frac{1}{\sqrt[3]{k}})^{N+1})$ ,

for sufficiently large $k$ , where $Tx\partial=\partial_{x}.$

3 Proof of TheoremThe non-normalized form of (2.15) is equal to

(3.1) $\lim_{Larrow\infty}\lim_{narrow\infty}(\frac{1}{\sqrt{2\pi}})^{L}\int_{R^{L}}e^{-\Sigma_{j=1}^{L}\frac{1}{2}(1-2i\sqrt[3]{k}x_{j}^{2}}na_{j})_{f^{L}(x_{L},x_{L-1},\cdots,x_{1})\prod_{j=1}^{L}dx_{j}},$

where $f^{L}(x_{L}, x_{L-1}, \cdots, x_{1})$ is a version of $f_{n}^{L}(x_{L}, x_{L-1}, \cdots, x_{1})$ such that

$\exp[-(\sum_{j=1}^{L}\sqrt{n}\beta_{j}|x_{j}|)^{2N+2}]$

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in $f_{n}^{L}(x)$ is replaced by

$\{\begin{array}{ll}\exp[-(\sum_{j=1}^{L}\sqrt{n}\beta_{j}x_{j})^{2N+2}], ifx_{j}\geq 0,\exp[-(-\sum_{j=1}^{L}\sqrt{n}\beta_{j}x_{j})^{2N+2}], ifx_{j}\leq 0.\end{array}$

When we consider the function $f^{L}(x_{N}, x_{L-1}, \cdots, x_{j}, \cdots, x_{1})$ as a function ofj-th coordinate $x_{j}$ , denote it by $f^{L}(x_{j})$ .

Since

$f^{L}(x_{j})= \hat{f}^{L}(0)+x_{j}\partial_{x_{j}}f^{L}(0)+\cdots+\frac{x_{j}^{2N}}{(2N)!}\partial_{x_{j}}^{2N}f^{L}(0)$

$+x_{j}^{2N+1} \int_{0}^{1}\frac{(1-\theta_{j})^{2N}}{(2N)!}\partial_{x_{j}}^{2N+1}\hat{f}^{L}(\theta_{j}x_{j})d\theta_{j},$

so that set

$Q_{j}^{0}f^{L}= \frac{1}{\sqrt{2\pi}}\int e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}\{\hat{f}^{L}(0)+x_{j}\partial_{x_{j}}\hat{f}^{L}(0)+\cdots+\frac{x_{j}^{2N}}{(2N)!}\partial_{x_{j}}^{2N}f^{L}(0)\}dx_{j}$

and

$Q_{j}^{1}f^{L}= \frac{1}{\sqrt{2\pi}}\int e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}x_{j}^{2N+1}(\int_{0^{1}}^{(1-\theta_{j})^{2N}}(2N)!\partial_{x_{j}}^{2N+1}\hat{f}^{L}(\theta_{j}x_{j})d\theta_{j})dx_{j}.$

Setting for any well differentiable function $g^{L}(x, x, \cdots, x_{2}, x_{1})$ ,

$D_{j}^{0}g^{L}= \frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}}\hat{g}^{L}(0)$,

$D_{j}^{m}g^{L}= \frac{1}{\sqrt{2\pi}}\int e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}x_{j}^{m}\partial_{x_{J}}^{m}\hat{g}^{L}(0)dx_{j}, 1\leq m\leq 2N,$

we get

$Q_{L}^{0}\cdots Q_{2}^{0}Q_{1}^{0}f^{L}$

(3.2)$=( \sum_{m=0}^{2N}D_{L}^{m})(\sum_{m=0}^{2N}D_{L-1}^{m})\cdots(\sum_{m=0}^{2N}D_{1}^{m}f^{L})$ .

Before proceeding to the estimate of the leading terms, we remark

Lemma 1. For any integer $0\leq m\leq 2N,$

$|D_{j}^{m}f^{L}| \leq\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j}}|}|\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}}|^{m}|\partial_{x_{j}}^{m}f^{L}(x_{L}, \cdots, x_{j+1},0, x_{j-1}, \cdots, x_{1})|.$

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Now we estimate the leading terms. The absolute value of the above (3.2)is dominated by

(3.3)

$|( \prod_{j=1}^{L}\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}})f^{L}(0,0, \cdots, 0)|+\sum_{1\leq j_{1}\leq L}(\sum_{m_{1}=1}^{2N}|D_{L}^{0}D_{L-1}^{0}\cdots D_{j_{1}}^{m_{1}}\cdots D_{2}^{0}D_{1}^{0}f^{L}|)$

$+ \sum_{1\leq j_{1}<j_{2}\leq L}(\sum_{m_{1},m_{2}=1}^{2N}|D_{L}^{0}D_{L-1}^{0}\cdots D_{j_{2}}^{m_{2}}\cdots D_{j_{1}}^{m_{1}}\cdots D_{2}^{0}D_{1}^{0}f^{L}|)$

$+\cdots\cdots+\cdots+\cdots$

$+ \sum_{1\leq j_{1}<j_{2}<j_{3}<j_{r}\leq L}\ldots(\sum_{\rangle}^{2..N}|D_{L}^{0}D_{L-1}^{0}\cdots D_{j_{r}}^{m_{r}}\cdots D_{j_{2}}^{m_{2}}\cdots D_{j_{1}}^{m_{1}}\cdots D_{2}^{0}D_{1}^{0}f^{L}|)$

$+\cdots\cdots+\cdots+\cdots$

By Lemma 1,

(3.4)$|D_{L}^{0}D_{L-1}^{0}\cdots D_{j_{r}}^{m_{f}}\cdots D_{j_{2}}^{m_{2}}\cdots D_{j_{1}}^{m_{1}}\cdots D_{2}^{0}D_{1}^{0}f^{L}|$

$\leq\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{L}}|}\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{L-1}}|}\ldots\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j_{r}}}|}|\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j_{r}}}}|^{m_{r}}$

$\cross\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j_{r}-1}}|}\ldots\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j_{r-1}}}|}|\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j_{r-1}}}}|^{m_{r-1}}$

$\cross\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j_{r-1}-1}}|,\backslash }\ldots\ldots\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j_{1}}}|}|\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j_{1}}}}|^{m_{1}}\ldots\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{1}}|}$

$\cross|(\partial_{x_{j_{r}}}^{m_{r}}\cdots\partial_{x_{j_{1}}}^{m_{1}}f^{L})(0,0, \cdots, 0,0, \cdots, 0,0)|.$

Here we state the key lemma will be omitted the proof because of therestriction of the pages.

Lemma 2. For any non-negative integer $N$ and $L$ , there exists some constants $A_{2N}\geq 0,$

$B_{2N}\geq 1$ and $\alpha(=2)\geq 1$ such that for any non-negative integers $1\leq j_{1}<j_{2}<\cdots<j_{r}\leq$

$L$ and $m_{j}\leq 2N,$

$|( \prod_{s=1}^{r}(\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j_{e}}}})^{\alpha_{j_{8}}})(\prod_{j=1.\cdots,L,\neq j_{1},j_{2_{\rangle}}\cdots,j_{r}}(\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}})^{m_{j}})|$

$\sup_{2N+1\leq\alpha_{j_{S}}\leq 4N+4,x_{j_{s}},1\leq s\leq r}|e^{-\Sigma_{s=1}^{r}\frac{1}{2}x_{j_{s}}^{2}}\partial_{x_{L}}^{m_{L}}\cdots\partial_{x_{j_{r}}}^{\alpha_{j_{r}}}$

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. . . $\partial_{x_{j_{2}}}^{\alpha_{j_{2}}}\cdots\partial_{x_{j_{1}}}^{\alpha_{j_{1}}}\cdots\partial_{x_{1}}^{m_{1}}f^{L}(0, \cdots, x_{j_{r}}, 0, \cdots, x_{j_{1}},0, \cdots, 0)|$

$\leq A_{2N}(\{(2N+1)_{j=1}.\cdots,\sum_{L,\neq j_{1},j_{2},\cdots,j_{r}}m_{j}+(2N+1)\sum_{s=1}^{r}\alpha_{j_{s}}\}!)^{\alpha}$

$( \prod_{s=1}^{r}(B_{2N}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{s}}|}})^{\alpha_{j_{\delta}}})$

$( \prod_{j=1,\cdots,L,\neq j_{1},j_{2},\cdots,j_{r}}(B_{2N}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}})^{m_{j}})$.

Then Lemma 2 implies

$|( \prod_{s=1}^{r}(\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j_{S}}}})^{m_{S}})(\partial_{x_{j_{r}}}^{m_{r}}\cdots\partial_{x_{j_{1}}}^{m_{1}}f^{L})(0,0, \cdots, 0,0, \cdots, 0,0)|$

$\leq A_{2N}(\{(2N+1)(\sum_{s=1}^{r}m_{s})\}!)^{\alpha}(\prod_{s=1}^{r}(B_{2N}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{s}}|}})^{m_{\epsilon}})$,

which, together with (3.4), yields

$|D_{L}^{0}D_{L-1}^{0}\cdots D_{j_{r}}^{m_{r}}\cdots D_{j_{2}}^{m_{2}}\cdots D_{j_{1}}^{m_{1}}\cdots D_{2}^{0}D_{1}^{0}f^{L}|$

(3.5)$\leq(\prod_{j=1}^{L}\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j}}|})A_{2N}(\{(2N+1)(\sum_{s=1}^{r}m_{s})\}!)^{\alpha}(\prod_{\epsilon=1}^{r}(B_{2N}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{s}}|}})^{m_{S}})$ .

Combining (3.3) and (3.5), we get that the absolute value of (3.2) is domi-nated by

(3.6)

$| \prod_{j=1}^{L}\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}}|\{A_{2N}+A_{2N}\sum_{1\leq j_{1}\leq L}(\sum_{m_{1}=1}^{2N}B_{2N}^{m_{1}}(\{(2N+1)m_{1}\}!)^{\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|^{m_{1}})$

$+A_{2N}$ $\sum$$1 \leq j_{1}<j_{2}\leq L(\sum_{m_{1},m_{2}=1}^{2N}B_{2N}^{m_{1}+m_{2}}(\{(2N+1)(m_{1}+m_{2})\}!)^{\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|^{m_{1}}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{2}}|}}|^{m_{2}})$

$+\cdots\cdots+\cdots\cdots$

$+A_{2N} \sum_{1\leq j_{1}<j_{2}<j_{3}<j_{r}\leq L}\ldots(\sum_{m_{1},m_{2},\cdots,m_{f}=1}^{2N}B_{2N}^{m_{1}+m_{2}+\cdot\cdot,+m_{r}}$

$( \{(2N+1)(m_{1}+m_{2}+\cdots+m_{r})\}!)^{\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|^{m_{1}}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{2}}|}}|m_{2}\cdots|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{r}}|}}|^{m_{r}})$

$+\cdots\cdots+\cdots\cdots$ .

18

Since

$(\{(2N+1)(m_{1}+m_{2}+\cdots+m_{r})\}!)^{\alpha}$

$\leq(\{r(2N+1)2N\}!)^{\alpha}\leq\{((p_{N})^{p_{N}})^{r}r!^{p_{N}}\}^{\alpha}$

$=((p_{N})^{p_{N}\alpha})^{r}r!^{p_{N}\alpha},$

where$p_{N}=(2N+1)2N,$

we have the above equation is dominated by

$| \prod_{j=1}^{L}\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}}|A_{2N}\{1+B_{2N}^{2N}\sum_{1\leq j_{1}\leq L}(\sum_{m_{1}=1}^{2N}((p_{N})^{p_{N}\alpha})|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|^{m_{1}})$

$+(B_{2N}^{2N})^{2}$ $\sum$$1 \leq j_{1}<j_{2}\leq L(\sum_{m_{1},m_{2}=1}^{2N}((p_{N})^{p_{N}\alpha})^{2}2!^{p_{N}\alpha}\frac{1}{1^{p_{N}\alpha}2^{p_{N}\alpha}}$

$j_{1}^{p_{N}\alpha}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|m_{1} j_{2}^{p_{N}\alpha}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{2}}|}}|^{m_{2}})$

$+\cdots\cdots+\cdots\cdots$

$+(B_{2N}^{2N})^{r} \sum_{1\leq j_{1}<j_{2}\triangleleft 3<j_{r}\leq L}\ldots(_{m,m_{2}}\sum_{1m_{r}=1}^{2..N},((p_{N})^{p_{N}\alpha})^{r}r!^{p_{N}\alpha}\frac{1}{1^{p_{N}\alpha}2^{p_{N}\alpha}\cdots r^{p_{N}\alpha}}$

$j_{1}^{p_{N}\alpha}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|m_{l} j_{2}^{p_{N}\alpha}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{2}}|}}|\begin{array}{ll}m_{2} \cdots j_{r}^{p_{N}\alpha}\end{array}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{r}}|}}|^{m_{r}})$

$+\cdots\backslash$ . . $+\cdots\cdots$

$+(B_{2N}^{2N})^{2}$ $\sum$$1 \leq j_{1}<j_{2}\leq L(\sum_{m_{1},m_{2}=1}^{2N}((p_{N})^{p_{N}\alpha})^{2}$

$j_{1}^{p_{N}\alpha}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|m_{l} j_{2}^{p_{N}\alpha}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{2}}|}}|^{m_{2}})$

$+\cdots\cdots+\cdots\cdots$

$+(B_{2N}^{2N})^{r} \sum_{1\leq j_{1}<j_{2}<j_{3}<j_{r}\leq L}\ldots(\sum_{m_{1},m_{2},\cdot\cdot m_{r}=1}^{2.N},((p_{N})^{p_{N}\alpha})^{r}j_{1}^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|^{m_{1}}$

19

$j_{2}^{p_{N}\alpha}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{2}}|}}|\begin{array}{ll}m2 \cdots j_{r}^{p_{N}\alpha}\end{array}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{\Gamma}}|}}|^{m_{r}})$

$+\cdots\cdots+\cdots\cdots$

$\leq|\prod_{j=1}^{L}\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}}|A_{2N}\{1+[B_{2N}^{3^{N}}(p_{N})^{p_{N}\alpha}]\sum_{1\leq j_{1}\leq L}(\sum_{m_{1}=1}^{2N}j_{1}^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|^{m_{1}})$

$+ \frac{1}{2!}[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}]^{2}(\sum_{j_{1}=1}^{L}(\sum_{m_{1}=1}^{2N}j_{1}^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|^{m_{1}}))^{2}$

$+\cdots\cdots+\cdots\cdots$

$+ \frac{1}{r!}[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}]^{r}(\sum_{j_{1}=1}^{L}(\sum_{m_{1}=1}^{2N}j_{1}^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|^{m_{1}}))^{r}$

$+\cdots\cdots+\cdots\cdots$ .

By the Assumption 1 such that

$\sum_{j=1}^{\infty}j^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|\leq c<+\infty,$

we have

$\sum_{j=1}^{L}(\sum_{m=1}^{2N}j^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|^{m})$

$\leq(\sum_{j=1}^{L}j^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|)(1+\sum_{j=1}^{L}j^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|)^{2N-1}$

$\leq c(1+c)^{2N-1}.$

Hence we get that the absolute value of (3.2) is dominated by

20

(3.7)

$| \prod_{j=1}^{L}\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}}|A_{2N}\{1+[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}](c(1+c)^{2N-1})$

$+ \frac{1}{2!}[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}]^{2}(c(1+c)^{2N-1})^{2}+\cdots\cdots+\cdots\cdots$

$+ \frac{1}{r!}[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}]^{r}(c(1+c)^{2N-1})^{r}+\cdots\cdots+\cdots\cdots$

$\leq|\prod_{j=1}^{L}\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}}|A_{2N}\exp[[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}]c(1+c)^{2N-1}]$

and we also have

(3.8)

$\sum_{r=2N+2}^{\infty}A_{2N}\frac{1}{r!}[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}]^{r}(\sum_{j=1}^{L}(\sum_{m=1}^{2N}j^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|^{m}))^{r}$

$\leq(\sum_{j=1}^{L}j^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|)^{2N+2}A_{2N}(1+c)^{(2N-1)(2N+1)}\exp[[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}](1+c)^{2N}].$

Further for any natural number $2\leq r\leq 2N+1$ , we get

(3.9)

$1 \leq j_{1}<j_{2}<j_{3<i_{r}\leq 1\leq m_{1},m_{2},\cdots,m_{r}\leq 2N+1,\Sigma_{s=1}^{r}}\sum_{L}\cdots(\sum_{m_{s}\geq 2N+2}A_{2N}B_{2N}^{m_{1}+m_{2}+\cdots+m_{r}}$

$( \{(2N+1)(m_{1}+m_{2}+\cdots+m_{r})\}!)^{\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|^{m_{1}}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{2}}|}}|m_{2}\cdots|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{r}}|}}|^{m_{r}})$

$\leq 1\leq j_{1<j_{2}<j_{3}<j_{r}\leq 1\leq m_{1},m_{2},\cdots,m_{r}\leq 2N+1,\Sigma_{\theta=1}^{r}}\sum_{L}\cdots(\sum_{m_{s}\geq 2N+2}$

$A_{2N}[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}]^{r}j_{1}^{p_{N}\alpha}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|m_{l} j_{2}^{p_{N}\alpha}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{2}}|}}|\begin{array}{ll}m_{2} \cdots j_{r}^{p_{N}\alpha}\end{array}| \frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{r}}|}}|^{m_{r}})$

$\leq A_{2N}[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}]^{2N+1}\sum_{1\leq j_{1}<j_{2}<j_{3}<j_{r}\leq L}\ldots(\sum_{1\leq m_{1},m_{2},\cdots,m_{r}\leq 2N+1,\Sigma_{\delta=1}^{r}m_{s}\geq 2N+2}$

$( \sum_{\dot{J}=1}^{L}j^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|)^{\Sigma_{s=i}^{r}m_{s}}$

$\leq(\sum_{j=1}^{\infty}j^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|)^{2N+2}A_{2N}[B_{2N}^{2N}(p_{N})^{p_{N}\alpha}]^{2N+1}(2N+1)^{2N+1}(1+c)^{(2N+1)^{2}}$

21

Since

$\partial_{x_{j}}^{m};\partial_{x_{j_{r-1}}}^{m_{r-1}}\cdots\partial_{x_{j_{1}}}^{m_{1}}\exp[-(\sum_{j=1}^{L}\sqrt{n}\beta_{j}|x_{j}|)^{2p+2}](0)=0$

if $m_{1}+\cdots+m_{r-1}+m_{r}\leq 2N+1$ , we have, together with (3.6), (3.8) and (3.9),

$Q_{L}^{0}\cdots Q_{2}^{0}Q_{1}^{0}f^{L}$

is rewritten as

(3.10)

$( \prod_{j=1}^{L}\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}})\{f^{L}(0,0, \cdots, 0)$

$+ \sum_{s=1}^{N}(\sum_{r=1}^{s}(\sum_{1\leq j_{1}<j_{2}<j_{3}<j_{r}\leq L}\ldots(\sum_{\rangle}\frac{1}{2^{m_{1}}m_{1}!(1-2in\sqrt[3]{k}a_{j\iota})^{m_{1}}}\partial_{x_{j_{1}}}^{2m_{1}}$

$\frac{1}{2^{m_{2}}m_{2}!(1-2in\sqrt[3]{k}a_{j_{2}})^{m_{2}}}\partial_{x_{j_{2}}}^{2m_{2}}\cdots\frac{1}{2^{m_{r}}m_{r}!(1-2in\sqrt[3]{k}a_{j_{r}})^{m_{r}}}\partial_{x_{jr}}^{2m_{r}}f^{L}(0,0, \cdots, 0))))$

$+O(( \sum_{j=1}^{\infty}j^{p_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|)^{2N+2})\}.$

Hence letting $Larrow\infty$ in (3.10), we have (2.16).

Now we estimate the remaining terms again by $t\grave{h}e$ method similar to thatin the leading terms. In this turn, we also begin with the proof of

Lemma 3. There eststs a positive constant $C_{N}$ such that

$|Q_{j}^{1}f^{L}| \leq\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j}}|}C_{N}(\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j}}|})^{\alpha_{j}}$

(3.11)$\sup e^{-\lrcorner^{x^{2}}}2|\partial_{x_{j}}^{\alpha_{j}}\hat{f}^{L}(x_{j})|.$

$2N+1\leq|\alpha_{j}|\leq 4N+4,x_{j}$

Proof. Apply the Fubini theorem and use the integration by parts formula, we have

$Q_{j}^{1}f^{L}= \int_{0}^{1}d\theta_{j}\frac{(1-\theta_{j})^{2N}1}{(2N)!\sqrt{2\pi}}\int e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}x_{j}^{2N+1}$

$\partial_{x_{j}}^{2N+1}\hat{f}^{L}(\theta_{j}x_{j})dx_{j}$

$= \int_{0}^{1}d\theta_{j}\frac{(1-\theta_{j})^{2N}1}{(2N)!\sqrt{2\pi}}\int\frac{-\partial_{x_{j}}}{1-2in\sqrt[3]{k}a_{j}}(e^{-\frac{1}{2}(1-2tn\sqrt[3]{k}a_{j})x_{j}^{2}})x_{j}^{2N}$

22

$\partial_{x_{j}}^{2N+1}\hat{f}^{L}(\theta_{j}x_{j})dx_{j}$

$= \int_{0}^{1}d\theta_{j}\frac{(1-\theta_{j})^{2N}1}{(2N)!(1-2in\sqrt[3]{k}a_{j})\sqrt{2\pi}}\int e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}2Nx_{j}^{2N-1}$

$\partial_{x_{j}}^{2N+1}\hat{f}^{L}(\theta_{j}x_{j})dx_{j}$

$+ \int_{0}^{1}d\theta_{j}\frac{(1-\theta_{j})^{2N}}{(2N)!}\frac{1}{(1-2in\sqrt[3]{k}a_{j})\sqrt{2\pi}}\int e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}x_{j}^{2N}$

$\theta_{j}\partial_{x_{j}}^{2N+2}f^{L}(\theta_{j}x_{j})dx_{j}.$

Repeating this process until $x_{j}$ vanishes, we have

$Q_{j}^{1}f^{L}= \int_{0}^{1}d\theta_{j}\frac{(1-\theta_{j})^{2N}1}{(2N)!(1-2in\sqrt[3]{k}a_{j})^{N+1\sqrt{2\pi}}}2N(2N-2)(2N-4)\cdots 2$

$\{-\partial_{x_{j}}^{2N+1}\hat{f}^{L}(0)+\int e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}$

(3.12)$\cross\theta_{j}\partial_{x_{j}}^{2N+2}\hat{f}^{L}(\theta_{j}x_{j})dx_{j}\}$

$+\cdots+\cdots\cdots$

$+. \int_{0}^{1}d\theta_{j}\frac{(1-\theta_{j})^{2N}}{(2N)!}\frac{1}{(1-2in\sqrt[3]{k}a_{j})^{2N+1\sqrt{2\pi}}}\int e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}$

$\theta_{j}^{2N+1}\partial_{x_{j}}^{4N+2}\hat{f}^{L}(\theta_{j}x_{j})dx_{j}.$

Setting

$M_{j}= \frac{1-x_{j}\partial_{x_{j}}}{1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}},$

we have

$M_{j}e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}=e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}.$

Noticing $M_{j}^{*}$ denotes the adjoint operator of $M_{j}$ in $L^{2}(dx_{j})$ , we get for $G(x)$ of being welldifferentiable function and of satisfying

$\lim e^{-\frac{1}{2}x_{j}^{2}}|G(x)|=0,$

$|x_{j}|arrow\infty$

$M_{j}^{*}G(x)= \frac{2G(x)}{(1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2})^{2}}+\frac{x_{j}\partial_{x_{j}}G(x)}{1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}},$

so that we have

23

(3.13)

$|(M_{j}^{*})^{2}G(x)| \leq\{|\frac{4G(x)}{1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}|+8|G(x)\Vert\frac{(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}{(1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2})^{4}}|$

$.+4| \frac{x_{j}}{1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}\Vert\frac{\partial_{x_{j}}G(x)}{(1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2})^{2}}|+|\frac{x_{j}}{1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}\Vert\frac{\partial_{x_{j}}G(x)}{1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}|$

$+2| \partial_{x_{j}}G(x)\Vert\frac{x_{j}}{1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}\Vert\frac{(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}{(1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2})^{2}}|$

$+| \frac{x_{j}^{2}}{1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}\Vert\frac{\partial_{x_{j}}^{2}G(x)}{1+(1-2in\sqrt{3}]ka_{j})x_{j}^{2}}|\}.$

Therefore for any positive integer $q$ , we have

$|e^{-\frac{1}{2}(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}(M_{j}^{*})^{2}(\partial_{x_{j}}^{q}\hat{f}^{L})(\theta_{j}x_{j})|$

$\leq 20(\sup_{|\alpha_{j}|\leq 2,x_{j},|\theta_{j}|\leq 1}(\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j}}|})^{\alpha_{j}}|e^{-\frac{1}{2}x_{j}^{2}}\partial_{x_{j}}^{\alpha_{j}+q}f^{L}(\theta_{j}x_{j})|)\cross|\frac{1}{1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}|.$

Notcing that

$\int|\frac{1}{1+(1-2in\sqrt[3]{k}a_{j})x_{j}^{2}}|dx_{j}\leq\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j}}|}\int\frac{1}{\sqrt{1+y^{4}}}dy$

and taking

$\sup_{2N+2\leq|\alpha_{j}|\leq 4N+4,x_{j},|\theta_{j}|\leq 1}(\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j}}|})^{\alpha_{j}}|e^{-\frac{1}{2}x_{j}^{2}}\partial_{x_{f}}^{\alpha_{j}}\hat{f}^{L}(\theta_{j}x_{j})|$

$\leq\sup_{2N+2\leq|\alpha_{j}|\leq 4N+4,x_{j},|\theta_{j}|\leq 1}(\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j}}|})^{\alpha_{j}}|e^{-\frac{1}{2}\theta_{j}^{2}x_{j}^{2}}\partial_{x_{j}^{j}}^{\alpha}f^{L}(\theta_{j}x_{j})|$

$\leq\sup_{2N+2\leq|\alpha_{j}|\leq 4N+4,x_{j}}(\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j}}|})^{\alpha_{j}}|e^{-\frac{1}{2}x_{j}^{2}}\partial_{x_{j}}^{\alpha_{j}}\hat{f}^{L}(x_{j})|$

and for example,

$| \frac{1}{(1-2in\sqrt[3]{k}a_{j})^{N+1\sqrt{2\pi}}}\{-\partial_{x_{j}}^{2N+1}\hat{f}^{L}(0)\}|$

$\leq|\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}}\Vert\frac{1}{\sqrt{1-2in\sqrt[3]{k}a_{j}}}|^{2N+1}\sup_{x_{j}}|e^{-\frac{1}{2}x_{j}^{2}}\partial_{x_{j}}^{2N+1}f^{L}(xj)|$

into account, we have the desired inequality from (3.12). $\square$

24

Now we return to the estimate the remaining terms such that

$| \sum_{k_{L},\cdots,k_{2},k_{1}=0,1,(k_{L},\cdot\cdot,k_{2},k_{1})\neq(0,0,\cdots,0)}.Q_{L}^{k_{L}}\cdots Q_{2}^{k_{2}}Q_{1}^{k_{1}}f^{L}|$

(3.14)

$\leq\sum_{r=1}^{L}\sum_{1\leq j_{1}<j_{2}<\cdot\cdot<j_{r}\leq L}.|Q_{L}^{0}\cdots Q_{j_{r}}^{1}\cdots Q_{j_{2}}^{1}\cdots Q_{j_{1}}^{1}\cdots Q_{1}^{0}f^{L}|.$

First we discuss the case of $r\geq 2.$

$|Q_{L}^{0}\cdots Q_{j_{r}}^{1}\cdots Q_{j_{2}}^{1}\cdots Q_{j_{1}}^{1}\cdots Q_{1}^{0}f^{L}|$

$=|( \sum_{m=0}^{2N}D_{L}^{m})\cdots\cdots Q_{j_{r}}^{1}\cdots Q_{j_{2}}^{1}\cdots Q_{j_{1}}^{1}\cdots(\sum_{m=0}^{2N}D_{1}^{m})f^{L}|$

$\leq|D_{L}^{0}\cdots\cdots Q_{j_{r}}^{1}\cdots Q_{j_{2}}^{1}\cdots Q_{j_{1}}^{1}\cdots D_{1}^{0}f^{L}|$

(3.15) $+\cdots+\cdots+\cdots$

$+( \sum_{1\leq l_{1}<l_{2}<\cdots<l_{r}\leq L,l_{1},l_{2},\cdots,l_{f}\neq j_{1},j_{2},\cdots,j_{r}})$

$( \sum_{m_{l_{1}},m_{l_{2}},\cdots,m_{l_{r}}=1,m_{i}=0,i\neq\{l_{1},l_{2},\cdots,l_{r}\}}^{2N})$

$|D_{L}^{m_{L}}\cdots Q_{j_{r}}^{1}\cdots Q_{j_{2}}^{1}\cdots Q_{j_{1}}^{1}\cdots D_{1}^{m_{1}}f^{L}|+\cdots+\cdots$

Next we estimate

(3.16) $|D_{L}^{m_{L}}\cdots D_{j_{r}+1}^{m_{jr+1}}Q_{j_{r}}^{1}D_{j_{r}-1}^{m_{jr-1}}\cdots Q_{j_{2}}^{1}\cdots Q_{j_{1}}^{1}\cdots D_{1}^{m_{1}}f^{L}|.$

By the manner similar to that in (3.4), we arrive at (3.16) is dominated by

$( \prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})(\prod_{J\neq j_{1},j_{2},j_{r}}\ldots,\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})m_{j}$

(3.17) $\cross C_{N}^{r}\sup_{2N+1\leq|\alpha_{j_{\theta}}|\leq 4N+4,x_{j_{s\rangle}}1\leq s\leq r}(\prod_{s=1}^{r}(\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})^{\alpha_{Js}})$

$\cross|e^{-\Sigma_{\epsilon=1}^{r}\frac{1}{2}x_{j_{s}}^{2}}(\partial_{x_{1}}^{m_{1}}\cdots\partial_{x_{j_{1}}^{j_{1}}}^{\alpha}\cdots\partial_{x_{j_{2}}}^{\alpha_{j_{2}}}\cdots\partial_{x_{j_{r}-1}}^{m_{j_{r}-1}}\partial_{x_{j_{r}}}^{\alpha_{j_{r}}}\partial_{x_{j_{r}+1}}^{m_{jr+1}}\cdots\partial_{x_{L}}^{m_{L}}$

$f^{L}((0, \cdots, 0, x_{j_{r}}, \cdots, x_{j_{2}}, \cdots, x_{j_{1}}, \cdots, 0)|.$

25

Suppose that

$1\leq l_{1}<l_{2}<\cdots<l_{t}\leq L, l_{1}, l_{2}, \cdots, l_{t}\neq j_{1},j_{2}, \cdots,j_{r}$

and

$m_{l_{j}}\geq 1, j=1,2, \cdots s, m_{i}=0, i\neq\{l_{1}, l_{2}, \cdots, l_{t}\}.$

Setting

$\hat{p}_{N}=(2N+1)(4N+4)$ ,

we have, by Lemma 2,

$( \prod_{j\neq j_{1},j_{2},j_{r}}\ldots,\frac{1}{|\sqrt{1-2in\sqrt[3]{k}a_{j}}|})m_{j}$

$\sup_{2N+1\leq|\alpha_{j_{\epsilon}}|\leq 4N+4,x_{j_{S}},1\leq s\leq r}(\prod_{s=1}^{r}(\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})^{\alpha_{j_{S}}})$

$\cross|e^{-\Sigma_{s=1}^{r}\frac{1}{2}x_{j_{S}}^{2}}(\partial_{x_{L}}^{m_{L}}\cdots\partial_{x_{jr}}^{\alpha_{j_{r}}}$

. . . $\partial_{x_{j_{2}}^{j_{2}}}^{\alpha}\cdots\partial_{x_{j_{1}}}^{\alpha_{j_{1}}}\cdots\partial_{x_{1}}^{m_{1}}f^{L}((0, \cdots, 0, x_{j_{r}}, \cdots, x_{j_{2}}, \cdots, x_{j_{1}}, \cdots, 0)|$

$\leq A_{2N}\sup_{2N+1\leq|\alpha_{j_{\delta}}|\leq 4N+4}(\{(2N+1)\sum_{s=1}^{r}\alpha_{j}. +(2N+1)\sum_{j=1}^{t}m_{l_{j}}\}!)^{\alpha}$

(3.18)$( \prod_{s=1}^{r}(B_{2N}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{\epsilon}}|}})^{\alpha_{Js}})(\prod_{j=1}^{t}(B_{2N}\frac{1}{1\sqrt{2\sqrt[3]{k}|\lambda_{j}|}})^{m_{1j}})$

$\leq A_{2N}\hat{p}_{N}^{\beta_{N}\alpha(t+r)}\{(t+r)!\}^{\hat{p}_{N}\alpha}$

1$\overline{1^{\hat{p}_{N}\alpha}2^{\hat{p}_{N}\alpha}\cdots(t+r)^{\hat{p}_{N}\alpha}}$

$\cross(\prod_{s=1}^{r}j_{s}^{\hat{p}_{N}\alpha}(B_{2N}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}})^{\alpha_{j_{\delta}}})$

$( \prod_{j=1}^{t}l_{j}^{\hat{p}_{N}\alpha}(B_{2N}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}})^{m_{t_{j}}})$ .

Summing up, (3.16),(3.17) and $(3.18),and$ setting $c_{N}= \max\{1,$ $C_{N}\hat{p}_{N}^{\hat{p}_{N}\alpha}B_{2N}^{4N+4}(1+$

$c)^{2N+3}\}$ , we have for sufficiently large $k,$

26

(3.19)

$|D_{L}^{m_{L}}D_{L-1}^{m_{L-1}}\cdots D_{j_{r}+1}^{m_{j_{r}+1}}Q_{j_{r}}^{1}D_{j_{r}-1}^{m_{j_{r}-1}}\cdots Q_{j_{2}}^{1}\cdots Q_{j_{1}}^{1}\cdots D_{1}^{m_{1}}f^{L}|$

$\leq(\prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})A_{2N}\hat{p}_{N}^{\hat{p}_{N}\alpha t}(\prod_{j=1}^{t}l_{j}^{\hat{p}_{N}\alpha}|B_{2N}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|^{m_{l_{j}}})$

$\cross C_{N}^{r}\hat{p}_{N}^{\hat{p}_{N}\alpha r}(\prod_{s=1}^{r}j_{s}^{\hat{p}_{N}\alpha}B_{2N}^{4N+4}(1+c)^{2N+3}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{\delta}}|}}|^{2N+1})$

$\leq(\prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})A_{2N}\hat{p}_{N}^{\hat{p}_{N}\alpha t}(\prod_{j=1}^{t}l_{j}^{\hat{p}_{N}\alpha}|c_{N}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|^{m_{t_{j}}})$

$\cross c_{N}^{r}(\prod_{s=1}^{r}j_{s}^{\hat{p}_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{s}}|}}|^{2N+1})$ .

Therefore by (3.19) and (3.15), we have

$|Q_{L}^{0}\cdots Q_{j_{r}}^{1}\cdots Q_{j_{2}}^{1}\cdots Q_{j_{1}}^{1}\cdots Q_{1}^{0}f^{L}|$

$\leq(\prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})c_{N}^{r}(\prod_{s=1}^{r}j_{s}^{\hat{p}_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{\epsilon}}|}}|^{2N+1})A_{2N}\{1+$

(3.20)$+ \hat{p}_{N}^{\hat{p}_{N}\alpha}(\sum_{1\leq l_{1}\leq L,l_{1}\neq j_{1},j_{2},\cdots,j_{r}})(\sum_{m_{l_{1}}=1,m_{i}=0,i\neq l_{1}}^{2N})l_{1}^{\hat{p}_{N}\alpha}|cN\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{l_{1}}|}}|^{m_{t_{1}}}$

$+\cdots+\cdots$

$+ \hat{p}_{N}^{\hat{p}_{N}\alpha t}(\sum_{1\leq\iota_{1<l_{2}<\cdots<l_{t}\leq L,l_{1},l_{2},\cdots,l_{t}\neq j_{1},j_{2},\cdots,j_{r}}})(\sum_{m_{l_{1}},m_{l_{2}},\cdots,m_{l_{t}}=1,m_{i}=0,i\neq\{\iota_{1},\iota_{2},,\iota_{t}\}}^{2N}\ldots)$

$( \prod_{j=1}^{t}l_{j}^{\hat{p}_{N}\alpha}|c_{N}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{l_{j}}|}}|^{m_{l_{j}}})+\cdots+\cdots\}.$

Since $m_{l_{t}}\leq 2N$ , setting

$M=A_{2N} \sum_{t=0}^{\infty}\frac{1}{t!}(c_{N}^{2N}\hat{p}_{N}^{\hat{p}_{N}\alpha})^{t}(\sum_{j=1}^{\infty}(\sum_{m=1}^{2N}j^{\hat{p}_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|^{m}))^{t}$

by the manner similar to that in estimating the leading terms, (3.20) is domi-nated by

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(3.21) $( \prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})Mc_{N}^{r}(\prod_{s=1}^{r}j_{S}^{\hat{p}_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{S}}|}}|^{2N+1})$.

Therefore by (3.14) and (3.21), we arrive at for sufficiently large $k,$

$| \sum_{k_{j}\geq 2}Q_{L}^{k_{L}}\cdots Q_{2}^{k_{2}}Q_{1}^{k_{1}}f^{L}|$

$\leq(\prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})\{\sum_{r=2}^{L}\frac{1}{r!}Mc_{N}^{r}(\sum_{j=1}^{L}j^{\hat{p}_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}|)^{(2N+1)r}\}$

$\leq(\prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})M(\sum_{j=1}^{L}j_{s}^{\hat{p}_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{\epsilon}}|}}|)^{(2N+2)}c_{N}^{2}c^{2N}\exp[c_{N}c^{2N+1}]$

$\leq(\prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})\{o((\frac{1}{\sqrt[3]{k}})^{N+1})\}.$

Next we discuss the case of $r=1$ . Since

$\partial_{x_{j}}^{2N+1}f^{L}(0)=0,$

so that the discussion after (3.12) implies there exists a constant $C$ such thatfor sufficiently large $k,$

$|D_{L}^{0}\cdots\cdots Q_{j_{1}}^{1}\cdots D_{1}^{0}f^{L}|$

$\leq C(\prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})(j_{1}^{\hat{p}_{N}\alpha}|\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}|)^{(2N+2)}$

Further, since

$|Q_{L}^{0}\cdots Q_{j_{1}}^{1}\cdots Q_{1}^{0}f^{N}-D_{L}^{0}\cdots Q_{j_{1}}^{1}\cdots D_{1}^{0}f^{L}|$

$\leq(\prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})[A_{2N}\sum_{r=1}^{\infty}\frac{1}{r!}(c_{N}^{2N}\hat{p}_{N}^{\beta_{N}\alpha})^{r}(\sum_{j=1}^{\infty}(\sum_{m=1}^{2N}j^{\hat{p}_{N}\alpha}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}m))^{r}]$

$\cross c_{N}j_{1}^{\hat{p}_{N}\alpha}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j_{1}}|}}2N+1$

and

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$( \sum_{j=1}^{\varphi}(\sum_{m=1}^{2N}j^{\hat{p}_{N}\alpha}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}}m))^{r})$

$\leq((\sum_{j=1}^{\infty}j^{\hat{p}_{N}\alpha}\frac{1}{\sqrt{2\sqrt[3]{k}|\lambda_{j}|}})(1+c)^{2N-1})^{r}$

we have for sufficiently large $k,$

$\sum_{j_{1}=1}^{L}|Q_{L}^{0}\cdots Q_{j_{1}}^{1}\cdots Q_{1}^{0}f^{L}|\leq(\prod_{j=1}^{L}\frac{1}{|\sqrt{1-2i\sqrt[3]{k}na_{j}}|})\{o((\frac{1}{\sqrt[3]{k}})^{N+1})\},$

which completes the proof.

References

[1] S. Albeverio and I. Mitoma , Asymptotic expansion of perturbativeChern-Simons theory via Wiener space, Bull. Sci. Math. 133 (2009),272-314.

[2] S. Axelrod and I. M. Singer, Chern-Simons perturbation theory, Pro-ceedings of the XXth International Conference on Differential GeometricMethods in Theoretical Physics, Vol. 1 (New York, 1991), 3-45, WorldSci. Publishing, River Edge, N.J., 1992.

[3] D. Bar-Natan and E. Witten, Perturbation expansion of Chern-Simonstheory with non-compact gauge group, Comm. Math. Phys. 141 (1991),423-440.

[4] K. T. Chen, Iterated path integrals, Bull. Amer. Soc. 83 (1977), 831-879.

[5] D. Fujiwara, The stationary phase method with an estimate of the re-mainder term on a space of large dimension, Nagoya Math. J. 124 (1991),61-97.

[6] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Math. Lecture Ser. 11, Publish or Perish, Inc.,Wilmington, Del., 1984.

[7] K. It\^o, Generalized uniform complex measures in the Hilbertian metricspace with their application to the Feynman integral, Proc. Fifth BerkeleySympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol.II, Contributions to Probability Theory, Part 1, 145-161, Univ. CaliforniaPress, Berkeley, Calif., 1967.

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[8] N. Kumano-go, Feynman path integrals as analysis on path space by timeslicing approximation, Bull. Sci. Math. 128 (2004), 197-251.

[9] I. Mitoma and S. Nishikawa, Asymptotic Expansion of the One LoopApproximation of the Chern-Simons Integral in an Abstract Wiener SpaceSetting, J. Funct. Anal. 253 (2007), 729-771.

[10] E. Witten, Quantum field theory and the Jones polynomial, Comm.Math. Phys. 121 (1989), 351-399.

Itaru MitomaDepartment of MathematicsSaga UniversitySaga 840-8502Japan$E$-mail address: mitomadms. saga-u. ac. jp

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